2 Assume that both x and y (where x 6= y) are solutions to the
consumer problem B(p, ω). By the convexity of the budget set B(p, ω) we have αx + (**1** − α)y ∈ B(p, ω) and, by the strict convexity of %, αx + (**1** − α)y ≻ z for all α ∈ (0, **1**) and z ∈ B(p, ω), which is a contradiction of x being % optimal in B(p, ω).

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(**1**) Write the payoff functions π **1** and π 2 (as a function of p **1** and p 2 ).
(2) Derive the best response function for each player. (3) Find the pure-strategy Nash equilibrium of this game.
(4) Derive the prices (p **1** , p 2 ) that maximize joint-profit, i.e., π **1** + π 2 .

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Introduction to Market Design and its Applications to School Choice.. Yosuke YASUDA.[r]

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(a) Show that the above data satisfy the Weak Axiom of revealed preference. (b) Show that this consumer’**s** behavior cannot be fully rationalized.
Hint: Assume there is some preference relation % that fully rationalizes the above data, and verify that % fails to satisfy transitivity.

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(a) Show that if u(x **1** , x 2 ) and v(x **1** , x 2 ) are both homogeneous of degree r, then
**s** (x **1** , x 2 ) := u(x **1** , x 2 ) + v(x **1** , x 2 ) is also homogeneous of degree r.
(b) Show that if u(x **1** , x 2 ) and v(x **1** , x 2 ) are quasi-concave, then m(x **1** , x 2 ) :=
min{u(x **1** , x 2 ), v(x **1** , x 2 )} is also quasi-concave.

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Solve the following problems in Snyder and Nicholson (11th):. 1.[r]

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【戦略】 個々**の**プレイヤーがとること**の**できる行動
【利得】 起こり得る行動**の**組み合わせに応じた満足度、効用
Q: ゲーム**の**解（予測）はどうやって与えられる？
A: 実はノイマン達は一般的な解を生み出せなかった…

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More on Roy’**s** Identity | もっとロア**の**恒等式
Roy’**s** identity says that the consumer’**s** Marshallian demand for good i is
simply the ratio of the partial derivatives of indirect utility with respect to p i
and ω after a sign change.

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“Soon after Nash ’s work, game-theoretic models began to be used in economic theory and political science,. and psychologists began studying how human subjects behave in experimental [r]

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A function f (x) is homothetic if f (x) = g(h(x)) where g is a strictly increasing function and h is a function which is homogeneous of degree **1**. Suppose preferences can be represented by a homothetic utility function. Then, show the followings.
(a) The marginal rate of substitution between any two goods depends only on the ratio of the demands consumed. That is M RS ij is identical whenever x x j i takes

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(b) Does this production function display increasing, constant, or decreasing re- turns to scale? Explain why.
(c) Formulate the cost minimization problem (you may denote a target output level by y). Then, solve it and derive the (minimum) cost function, c(w **1** , w 2 , y).

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Suppose that the decision maker’**s** preferences under uncertainty are described by the vNM utility function, u(x) = √ x.
(a) Is the decision maker risk-averse, risk-neutral, or risk-loving? Explain why. (b) Calculate the absolute risk aversion and the relative risk aversion, respectively.

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6. General Equilibrium (30 points)
Consider a production economy with two individuals, Ann (A) and Bob (B), and two goods, leisure x **1** and a consumption good x 2 . Ann and Bob have equal en- dowments of time (= ω **1** ) to be allocated between leisure and work, so the total

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(a) Suppose % is represented by utility function u(·). Then, u(·) is quasi-concave IF AND ONLY IF % is convex.
(b) Marshallian demand function is ALWAYS weakly decreasing in its own price. (c) Lagrange’**s** method ALWAYS derives optimal solutions for any optimization

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vNM Utility Function (**1**)
Note the function U is a utility function representing the preferences on L(S) while v is a utility function defined over S, which is the building block for the construction of U (p). We refer to v as a vNM (Von Neumann-Morgenstern) utility function.

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not have any Nash equilibrium, including mixed strategy equilibrium. **5**. Question **5** (6 points, Review)
A crime is observed by a group of n people. Each person would like the police to be informed but prefers that someone else make the phone call. They choose either “call” or “not” independently and simultaneously. A person receives 0 payoff if no one calls. If someone (including herself) makes a call, she receives v while making a call costs c. We assume v > c so that each person has an incentive to call if no one else will call.

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Consider a consumer problem. Suppose that a choice function x(p; !) satis…es Walras’**s** law and WA. Then, show that x(p; !) is homogeneous of degree zero. 6. Lagrange’**s** Method
You have two …nal exams upcoming, Mathematics (M) and Japanese (J), and have to decide how to allocate your time to study each subject. After eating, sleeping, exercising, and maintaining some human contact, you will have T hours each day in which to study for your exams. You have …gured out that your grade point average (G) from your two courses takes the form

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with x = (y, z) where y is a scalar, z is an n-dimensional consumption vector, and V (·) is a real valued function. The consumption set X = R n +**1**
+ .
(a) Show that if V is concave, U is quasi-concave. (b) Show that if U is quasi-concave, V is concave. **5**. Question **5** (4 points)

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Solve the following problems in Snyder and Nicholson (11th):. 1.[r]

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